We study BDD-based bucket elimination, an approach to satisfiability testing using variable elimination which has seen several practical implementations in the past. We prove that it allows solving the standard pigeonhole principle formulas efficiently, when allowing different orders for variable elimination and BDD-representations, a variant of bucket elimination that was recently introduced. Furthermore, we show that this upper bound is somewhat brittle as for formulas which we get from the pigeonhole principle by restriction, i.e., fixing some of the variables, the same approach with the same variable orders has exponential runtime. We also show that the more common implementation of bucket elimination using the same order for variable elimination and the BDDs has exponential runtime for the pigeonhole principle when using either of the two orders from our upper bound, which suggests that the combination of both is the key to efficiency in the setting.

Probabilistic graphical models have emerged as a powerful modeling tool for several real-world scenarios where one needs to reason under uncertainty. A graphical model's partition function is a central quantity of interest, and its computation is key to several probabilistic reasoning tasks. Given the #P-hardness of computing the partition function, several techniques have been proposed over the years with varying guarantees on the quality of estimates and their runtime behavior. This paper seeks to present a survey of 18 techniques and a rigorous empirical study of their behavior across an extensive set of benchmarks. Our empirical study draws up a surprising observation: exact techniques are as efficient as the approximate ones, and therefore, we conclude with an optimistic view of opportunities for the design of approximate techniques with enhanced scalability. Motivated by the observation of an order of magnitude difference between the Virtual Best Solver and the best performing tool, we envision an exciting line of research focused on the development of portfolio solvers.

We compute exact literal-weighted model counts of CNF formulas. Our algorithm employs dynamic programming, with Algebraic Decision Diagrams as the primary data structure. This technique is implemented in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We also compare ADDMC to state-of-the-art exact model counters (Cachet, c2d, d4, miniC2D, and sharpSAT) on the two largest CNF model counting benchmark families (BayesNet and Planning). ADDMC solves the most benchmarks in total within the given timeout.

A weighted constraint satisfaction problem (WCSP) is a constraint satisfaction problem in which preferences among solutions can be expressed. Bucket elimination is a complete technique commonly used to solve this kind of constraint satisfaction problem. When the memory required to apply bucket elimination is too high, a heuristic method based on it (denominated mini-buckets) can be used to calculate bounds for the optimal solution. Nevertheless, the curse of dimensionality makes these techniques impractical on large scale problems. In response to this situation, we present a memetic algorithm for WCSPs in which bucket elimination is used as a mechanism for recombining solutions, providing the best possible child from the parental set. Subsequently, a multi-level model in which this exact/metaheuristic hybrid is further hybridized with branch-and-bound techniques and mini-buckets is studied. As a case study, we have applied these algorithms to the resolution of the maximum density still life problem, a hard constraint optimization problem based on Conways game of life. The resulting algorithm consistently finds optimal patterns for up to date solved instances in less time than current approaches. Moreover, it is shown that this proposal provides new best known solutions for very large instances.

A weighted constraint satisfaction problem (WCSP) is a constraint satisfaction problem in which preferences among solutions can be expressed. Bucket elimination is a complete technique commonly used to solve this kind of constraint satisfaction problem. When the memory required to apply bucket elimination is too high, a heuristic method based on it (denominated mini-buckets) can be used to calculate bounds for the optimal solution. Nevertheless, the curse of dimensionality makes these techniques impractical on large scale problems. In response to this situation, we present a memetic algorithm for WCSPs in which bucket elimination is used as a mechanism for recombining solutions, providing the best possible child from the parental set. Subsequently, a multi-level model in which this exact/metaheuristic hybrid is further hybridized with branch-and-bound techniques and mini-buckets is studied. As a case study, we have applied these algorithms to the resolution of the maximum density still life problem, a hard constraint optimization problem based on Conway's game of life. The resulting algorithm consistently finds optimal patterns for up to date solved instances in less time than current approaches. Moreover, it is shown that this proposal provides new best known solutions for very large instances.